When \(V\)  is a finite dimensional vector space over the field \(K\)  the notions of
\(K\) -linear relations and the linear relations of Boreale coincide. Consider a linear
relation (as defined in [1]}) \(R\)  over the vector space
\(V=K(X)\)  with basis \(X\)  where \(K\)  is a field. By definition, there exists a
subspace \(U\)  of the vector space \(V\)  over \(K\)  such that \(uRv\)  iff \(u-v \in U\) .
It is easy to check that the equivalence relation \(R=R_U=\lbrace (u,v) \mid u-v \in U\rbrace \)  is a congruence. Then consider the canonical linear map \(f_U: V \rightarrow V/R_U\) 
to the quotient space which maps an element \(w \in V\)  to \([w]_R\) . Now \(w \in U\) 
iff \(w = u-v\)  for some \(u,v \in V\)  with \(uRv\) . Therefore, \(f_U(w)=f_U(u-v)=f_U(u) - f_U(v) = 0\)  since \([u]=[v]\) , i.e., \(f_U\)  sends all elements in
\(U\)  to 0. It follows that \(u-v \in U\)  iff \(f_U(u)=f_U(v)\)  and thus \(R\)  is a
\(K\) -linear relation. Conversely, if \(R\)  is a \(K\) -linear relation over \(V\)  then \(R=R_f\)  for
some \(f : V \rightarrow W\) . Let \(U = \lbrace w \mid f(w)=0\rbrace \) . Clearly \(U\)  is a subspace of
\(V\)  and \(u -v \in U\)  iff \(f(u)=f(v)\) , i.e., \(R\)  is a linear relation. In
addition, when \(V\)  is a vector space over a field \(K\)  it is routine to verify
that the notions of a linear weighted bisimulation as in
[1]}, [3]} and a \(K\) -linear bisimulation
coincide, as the two definitions are identical. Hence, \(K\) -linear bisimulation
is a more general notion.
